Dynamical Algebraic Combinatorics and Webs

Many combinatorial objects with beautiful enumerative formulae also have remarkable dynamical behavior and underlying algebraic structure; dynamical algebraic combinatorics studies such objects and actions. Webs are combinatorial diagrams that encode polynomial invariants. They are a powerful tool in combinatorial representation theory and yield novel connections between diverse areas. The PI, jointly with collaborators, recently obtained a rotation-invariant sl4-web basis, which had been sought for almost 30 years. This project will extend this work to higher rank webs using dynamical algebraic combinatorics as a guide, as well as advance understanding of alternating sign matrices and plane partitions, which arose naturally (and surprisingly!) in this study of sl4-webs. This work also includes mentoring students and early career researchers and outreach to K-12 students. This research is comprised of several projects. A primary aim is to build on the sl4 web basis discovery described above to develop such an sl5-web basis, in a way that provides insight towards a long-sought complete solution for arbitrary rank. Another project studies promotion and rowmotion, fundamental actions in dynamical algebraic combinatorics, on more general objects and lattices. Alternating sign matrices are intriguing combinatorial objects related to the square ice model of statistical physics. It has been a long-standing open problem to construct an explicit bijection between these and totally symmetric self-complementary plane partitions. A third project builds on the PI’s recent work with Huang that interprets each of these objects as different types of pipe dreams, which give combinatorial expansions of Schubert polynomials, to find the largest known partial bijection. It also explores the interpretations of these objects and their generalizations as webs. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.